A composite function can be written as $w\bigl(u(x)\bigr)$, where $u$ and $w$ are basic functions. Is $f(x)=\tan^4(x)$ a composite function? If so, what are $u$ and $w$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $f$ is composite. $u(x)=\tan(x)$ and $w(x)=x^4$. (Choice B) B $f$ is composite. $u(x)=x^4$ and $w(x)=\tan(x)$. (Choice C) C $f$ is not a composite function.
Explanation: Composite and combined functions A composite function is where we make the output from one function, in this case $u$, the input for another function, in this case $w$. We can also combine functions using arithmetic operations, but such a combination is not considered a composite function. The inner function The inner function is the part we evaluate first. Frequently, we can identify the correct expression because it will appear within a grouping symbol one or more times in our composed function. Here, we only have $x$ inside of the parentheses. However, we can rewrite our expression as $(\tan(x))^4$. Now we have $\tan(x)$ inside a grouping symbol. We evaluate this expression first, so $u(x)=\tan(x)$ is the inner function. The outer function Then we raise the entire output of $u$ to the power of $4$. So $w(x)=x^4$ is the outer function. Answer $f$ is composite. $u(x)=\tan(x)$ and $w(x)=x^4$. Note that there are other valid ways to decompose $f$, especially into more complicated functions.